Recently, I have been reading some algebra. This has been immensely enjoyable; I had forgotten how much I used to like algebra. The material here is by no means advanced, but relies on some basic definitions from group theory.
Suppose I have two groups H and K. Can we combine them together? I learnt some years ago now that there is a straightforward way of forming a ‘sum’ of these two objects: One takes the cartesian product HK, and bestows it with a group operation in the most basic and obvious way possible: (h,k)(h’,k’) = (hh’,kk’). The group that results is known as the direct product of H and K.
The same can be done with vector spaces: Given two vector spaces V and W, I can form the direct sum of them, which is thought of as the vector space consisting of ordered pairs of the form (v,w) with v in V and w in W. It is easy to guess what the rules for addition and scalar multiplication must be. However, soon after learning these two definitions, I began to realise that at least a small amount of wool was obscuring my eyes: Suppose I have a vector space X and two subspaces Y and Z such that every vector x in X can be written uniquely in the form x = y + z, for y in Y and z in Z. I would then be invited to observe how this means that X was the direct sum of Y and Z. This didn’t sit well with me. It clearly wasn’t quite the same thing: There was no cartesian product; there was no way that x = y + z was the same as x = (y,z).
I soon came across the same disparity in group theory as well:One may have a group G that can be written as G = HK for two normal subgroups H and K (this means, as in the vector space example, that every g in G can be expressed uniquely as g = hk for h in H and k in K). In what sense is G the direct product of H and K?
There are really two different kinds of direct product, as you have now seen. For me, only when trying to fully understand the more complicated semidirect product of groups did the difference become stark enough for me to really take notice. I’ll now stick with groups rather than switching between groups and vector spaces.
Let’s return to the two groups H and K, which I started with. They are completely separate groups with separate identity elements and group operations etc. Let’s denote their direct product by G. The first mathematical observation I’d like to make is that the group G contains copies of both H and K as normal subgroups: The subgroup H’ of G consisting of elements of the form (h,1) is isomorphic to H and the normality is obvious: For x,h in H and k in K, we have
We define K‘ analogously. Secondly, it is trivial to observe that (by construction) every element g of G can be expressed in the form (h,k) = (h,1)(1,k) for h in H and k in K. We write this fact as G = H’K’. Thirdly, the only element of G which belongs to both of H’ and K’ is the identity element (1,1).
The key conceptual point about direct products is that the copies of H and K inside G stay separate. They do not interact with each other at all. The structure of G is determined by H interacting with itself, in the form of the normal subgroup H’, and, simultaneously but independently, K interacting with itself, in the form of the normal subgroup K’.
The construction we have just described is called the external direct product of H and K. The three observations which we made are the core of what a direct product of groups is, because it is from these observations that one may see both external and internal direct products, which we now go on to discuss, in the same light.
Let us suppose now that we start with a (completely different) group G, and normal subgroups H’ and K’. Suppose further that every element g of G can be written as g = hk for h in H’ and k in K’. Lastly, suppose that the only element of G which belongs to both H’ and K’ is the identity element. In this case, we say that G is the internal direct product of H’ and K’. Totally painless. If you ask me, this is the more natural concept.
I was introduced to these concepts in the order I have exposited them here: The external direct product first, and then the internal direct product. The external direct product seemed obvious and natural at first: A robust and basic way to build new groups from old. However, since then, my point of view has completely changed. I think that the other way around is the more natural order.
The internal direct product captures the following situation: You are given a group G, that, on closer inspection is built out of two groups H, K, in such a way that they do not interact with each other – it is easy to show that every element g of G can be written uniquely in the form g=hk, and that the elements of H and K commute with each other (this is even easier to see by looking at the external direct product) . The summands H and K really are like separate dimensions of G – think of the two ‘coordinates’ of under addition. Now, suppose instead that you are given two separate groups H and K. The concept of the external direct product answers the following question: Is there a group which contains both H and K (or rather isomorphic copies thereof) as subgroups and which is the internal direct product of said subgroups? The answer is yes and the external direct product is exactly this group (‘up to isomorphism’, obviously). The interpretation I am driving home here is that the external direct product is not nice and natural but rather a contrived and somewhat cumbersome construction used to recreate and achieve the more natural scenario of the internal direct product.
In the next post we will start to think about what happens when there is some interaction between the two subgroups which make up a group in this way and thus explore the concept of the semi-direct product.